if it's a regular room with square walls run two pieces of string. one from opposing corners top to bottom across the room. and the other the same in different corners. where they cross is the centre.
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You can use plumb lines to find its center of gravity. Hold it from one point and draw vertical line in direction of gravity (plumb line). Hold it from another point and do the same, and a third point if necessary; where the lines cross is center of gravity
Draw a cord through the circle (a line through the circle, but not too close to where you imagine the center to be). With construction techniques, find the perpendicular at the center point of the cord, and draw the perpendicular. Do the same thing again starting with a different cord, and the two perpendiculars will intersect at the center of the circle.
This will work with any rectangle, but not with some other weird shape: -- Stretch a string on the diagonal, i.e. from one corner to the opposite corner. Tape it down. -- Stretch another string on the other diagonal, i.e. between the two remaining corners. The point where the second string crosses the first string is the center of the floor.